Let $(\Omega ,\mathcal F,\mathbb P)$ a measure space. I know that a random variable $X:\Omega \to \mathbb R$ define a measure $\nu$ on $(\mathbb R,\mathcal B(\mathbb R))$ by $$\nu(A)=\mathbb P\{X\in A\}.$$ This is called the pushforward measure.
But it's written in my lecture that it also define a measure on $\Omega $. How is this possible ? What is the measure induced by $X$ on $\Omega $ ? Is is the pullback measure ?